Problem: Which of the definite integrals is equivalent to the following limit? $ \lim_{n\to\infty} \sum_{i=1}^n 4\cdot\dfrac5n$ Choose 1 answer: Choose 1 answer: (Choice A) A $ \int_0^{4} 5x\,dx$ (Choice B) B $ \int_0^{5} 4\,dx$ (Choice C) C $ \int_0^{5} 4x\,dx$ (Choice D) D $ \int_0^{4} 5\,dx$
Solution: Notice that the terms of the summation $\sum_{i=1}^n 4\cdot\dfrac5n$ do not depend on the index of summation $i$. This means we can easily evaluate the sum and actually find its limit. What is the value of the sum? The terms of the sum simplify to $20/n$. When we add $20/n$ to itself $n$ times, we get $\dfrac{20}n\cdot n=20$. This shows that $\sum_{i=1}^n 4\cdot\dfrac5n=20$. What is the limit of the sum as $n\to\infty$ ? $\lim_{n\to\infty} \sum_{i=1}^n 4\cdot\dfrac5n = \lim_{n\to\infty} 20 = 20$ Do any of the given integrals have a value of $20$ ? Instead of evaluating all three integrals, let's try to get a hint from the summation $\sum_{i=1}^n 4\cdot\dfrac{5}n\,$. It looks like a Riemann sum for a definite integral of the constant function $4$. The Riemann sum uses $n$ subintervals of equal width $5/n$. This means the interval of integration $[a,b]$ has width $b-a=n\cdot\dfrac5n=5$. Do any of the given integrals have an integrand of $4$ and an interval of width $5$ ? The integral $ \int_0^{5} 4\,dx$ integrates the constant function $4$ over an interval of width $5$. Moreover, $ \int_0^{5} 4\,dx = 4\cdot(5-0)=20$. Thus, the limit of the Riemann sum and the value of the integral both equal $20$. The correct answer is $ \lim_{n\to\infty}\sum_{i=1}^n 4\cdot\dfrac{5}n = \int_0^{5} 4\,dx$.